Elliptic K3 Surfaces with Abelian and Dihedral Groups of Symplectic Automorphisms

We analyze K3 surfaces admitting an elliptic fibration ? and a finite group G of symplectic automorphisms preserving this elliptic fibration. We construct the quotient elliptic fibration ?/G comparing its properties to the ones of ?.

We show that if ? admits an n-torsion section, its quotient by the group of automorphisms induced by this section admits again an n-torsion section, and we describe the coarse moduli space of K3 surfaces with a given finite group contained in the Mordell–Weil group.

Considering automorphisms coming from the base of the fibration, we find the Mordell–Weil lattice of a fibration described by Kloosterman, and we find K3 surfaces with dihedral groups as group of symplectic automorphisms. We prove the isometries between lattices described by the author and Sarti and lattices described by Shioda and by Greiss and Lam.     

On base manifolds of Lagrangian fibrations

We consider base spaces of Lagrangian fibrations from singular symplectic varieties.After defining cohomologically irreducible symplectic varieties,we construct an example of Lagrangian fibration whose base space is isomorphic to a quotient of the projective space.We also prove that the base space of Lagrangian fibration from a cohomologically symplectic variety is isomorphic to the projective space provided that the base space is smooth.     

On extensions of a symplectic class

Let F be a fibration on a simply-connected base with symplectic fiber (M,ω). Assume that the fiber is nilpotent and T2k-separable for some integer k or a nilmanifold. Then our main theorem, Theorem 1.8, gives a necessary and sufficient condition for the cohomology class [ω] to extend to a cohomology class of the total space of F. This allows us to describe Thurston?s criterion for a symplectic fibration to admit a compatible symplectic form in terms of the classifying map for the underlying fibration. The obstruction due to Lalond and McDuff for a symplectic bundle to be Hamiltonian is also rephrased in the same vein. Furthermore, with the aid of the main theorem, we discuss a global nature of the set of the homotopy equivalence classes of fibrations with symplectic fiber in which the class [ω] is extendable.     

Deformation quantization of pseudo-symplectic (Poisson) groupoids

We introduce a new kind of groupoid—a pseudo-étale groupoid, which provides many interesting examples of noncommutative Poisson algebras as defined by Block, Getzler, and Xu. Following the idea that symplectic and Poisson geometries are the semiclassical limits of the corresponding quantum geometries, we quantize these noncommutative Poisson algebras in the framework of deformation quantization. Received: September 2004 Revision: September 2005 Accepted: September 2005 Dedicated to A. Weinstein on his 60th birthday     

A weak version of the Lipman–Zariski conjecture

Markushevich and Tikhomirov provided a construction of an irreducible symplectic V-manifold of dimension 4, the relative compactified Prym variety of a family of curves with involution, which is a Lagrangian fibration with polarization of type (1,2). We give a characterization of the dual Lagrangian fibration. We also identify the moduli space of Lagrangian fibrations of this type and show that the duality defines a rational involution on it.     

Partial Dirac cohomology and tempered representations

The tempered representations of a real reductive Lie group G are naturally partitioned into series associated with conjugacy classes of Cartan subgroups H of G. We define partial Dirac cohomology, apply it for geometric construction of various models of these H–series representations, and show how this construction fits into the framework of geometric quantization and symplectic reduction.     

Restrictions on symplectic fibrations

Moduli spaces of stable pseudoholomorphic curves can be defined parametrically, i.e., over total spaces of symplectic fibrations. This imposes several restrictions on the spectral sequence of a symplectic fibration. We prove, among others, that under certain assumptions the spectral sequence collapses at E₂. In the appendix, we prove nontriviality of certain Gromov-Witten invariant for blow-ups. As an application we obtain that any Hamiltonian fibration with the blow-up of  along four dimensional submanifold as a fibre c-splits. That is its spectral sequence collapses.     

Star Product for Second-Class Constraint Systems from a BRST Theory

We propose an explicit construction of the deformation quantization of a general second-class constraint system that is covariant with respect to local coordinates on the phase space. The approach is based on constructing the effective first-class constraint (gauge) system equivalent to the original second-class constraint system and can also be understood as a far-reaching generalization of the Fedosov quantization. The effective gauge system is quantized by the BFV–BRST procedure. The star product for the Dirac bracket is explicitly constructed as the quantum multiplication of BRST observables. We introduce and explicitly construct a Dirac bracket counterpart of the symplectic connection, called the Dirac connection. We identify a particular star product associated with the Dirac connection for which the constraints are in the center of the respective star-commutator algebra. It is shown that when reduced to the constraint surface, this star product is a Fedosov star product on the constraint surface considered as a symplectic manifold.     

Cyclic cocycles on deformation quantizations and higher index theorems

We construct a nontrivial cyclic cocycle on the Weyl algebra of a symplectic vector space. Using this cyclic cocycle we construct an explicit, local, quasi-isomorphism from the complex of differential forms on a symplectic manifold to the complex of cyclic cochains of any formal deformation quantization thereof. We give a new proof of Nest-Tsygan's algebraic higher index theorem by computing the pairing between such cyclic cocycles and the K-theory of the formal deformation quantization. Furthermore, we extend this approach to derive an algebraic higher index theorem on a symplectic orbifold. As an application, we obtain the analytic higher index theorem of Connes-Moscovici and its extension to orbifolds.     

Hyperkähler Syz Conjecture and Semipositive Line Bundles

Let M be a compact, holomorphic symplectic Kähler manifold, and L a non-trivial line bundle admitting a metric of semipositive curvature. We show that some power of L is effective. This result is related to the hyperkähler SYZ conjecture, which states that such a manifold admits a holomorphic Lagrangian fibration, if L is not big.     

An algebraic index theorem for orbifolds

Using the concept of a twisted trace density on a cyclic groupoid, a trace is constructed on a formal deformation quantization of a symplectic orbifold. An algebraic index theorem for orbifolds follows as a consequence of a local Riemann-Roch theorem for such densities. In the case of a reduced orbifold, this proves a conjecture by Fedosov, Schulze, and Tarkhanov. Finally, it is shown how the Kawasaki index theorem for elliptic operators on orbifolds follows from this algebraic index theorem.     

Fedosov quantization in positive characteristic

We study the problem of deformation quantization for (algebraic) symplectic manifolds over a base field of positive characteristic. We prove a reasonably complete classification theorem for one class of such quantizations; in the course of doing it, we also introduce a notion of a restricted Poisson algebra - the Poisson analog of the standard notion of a restricted Lie algebra - and we prove a version of the Darboux Theorem valid in the positive characteristic setting.

     

Quantization of the Geodesic Flow on Quaternion Projective Spaces

We study a problem of the geometric quantization for the quaternionprojective space. First we explain a Kähler structure on the punctured cotangent bundleof the quaternion projective space, whose Kähler form coincides withthe natural symplectic form on the cotangent bundle and show thatthe canonical line bundle of this complex structure is holomorphicallytrivial by explicitly constructing a nowhere vanishing holomorphicglobal section. Then we construct a Hilbert space consisting of acertain class of holomorphic functions on the punctured cotangentbundle by the method ofpairing polarization and incidentally we construct an operatorfrom this Hilbert space to the L ₂ space of the quaternionprojective space. Also we construct a similar operator between thesetwo Hilbert spaces through the Hopf fiberation.We prove that these operators quantizethe geodesic flow of the quaternion projective space tothe one parameter group of the unitary Fourier integral operatorsgenerated by the square root of the Laplacian plus suitable constant.Finally we remark that the Hilbert space above has the reproducing kernel.     

Symplectic Implosion

Let K be a compact Lie group. We introduce the process of symplectic implosion, which associates to every Hamiltonian K-manifold a stratified space called the imploded cross-section. It bears a resemblance to symplectic reduction, but instead of quotienting by the entire group, it cuts the symmetries down to a maximal torus of K. We examine the nature of the singularities and describe in detail the imploded cross-section of the cotangent bundle of K, which turns out to be identical to an affine variety studied by Gelfand, Popov, Vinberg, and others. Finally we show that "quantization commutes with implosion".     

Harmonic symplectic spinors on Riemann surfaces

Symplectic spinor fields were considered already in the 70th in order to give the construction of half-densities in the context of geometric quantization. We introduced symplectic Dirac operators acting on symplectic spinor fields and started a systematical investigation. In this paper, we motivate the notion of harmonic symplectic spinor fields. We describe how many linearly independent harmonic symplectic spinors each Riemann surface admits. Furthermore, we calculate the spectrum of the symplectic spinor Laplacian on the complex projective space of complex dimension 1.     

Quasimodes and Bohr-Sommerfeld Conditions for the Toeplitz Operators

Abstract

This article is devoted to the quantization of the Lagrangian submanifolds in the context of geometric quantization. The objects we define are similar to the Lagrangian distributions of the cotangent phase space theory. We apply this to construct quasimodes for the Toeplitz operators and we state the Bohr-Sommerfeld conditions under the usual regularity assumption. To compare with the Bohr-Sommerfeld conditions for a pseudodifferential operator with small parameter, the Maslov index, defined from the vertical polarization, is replaced with a curvature integral, defined from the complex polarization. We also consider the quantization of the symplectomorphisms, the realization of semi-classical equivalence between two different quantizations of a symplectic manifold and the microlocal equivalences.     

Toeplitz Operators on Symplectic Manifolds

We study the Berezin-Toeplitz quantization on symplectic manifolds making use of the full off-diagonal asymptotic expansion of the Bergman kernel. We give also a characterization of Toeplitz operators in terms of their asymptotic expansion. The semi-classical limit properties of the Berezin-Toeplitz quantization for non-compact manifolds and orbifolds are also established. Second-named author partially supported by the SFB/TR 12.     

Families of Lagrangian fibrations on hyperkähler manifolds

A holomorphic Lagrangian fibration on a holomorphically symplectic manifold is a holomorphic map with Lagrangian fibers. It is known (due to Huybrechts) that a given compact manifold admits only finitely many holomorphic symplectic structures, up to deformation. We prove that a given compact, simple hyperkähler manifold with _b2?7$b_2?7$ admits only finitely many deformation types of holomorphic Lagrangian fibrations. We also prove that all known hyperkähler manifolds are never Kobayashi hyperbolic.